Relativistic Frames of Reference

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Relativistic Frames of Reference

From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that

Figure 2.1: Primed reference frame moving in the z direction with velocity v.

[math]\displaystyle{ t= t' }[/math] [math]\displaystyle{ x=x' }[/math] [math]\displaystyle{ y=y' }[/math] [math]\displaystyle{ z=z'+vt }[/math]

Using Einstein's Theory of Relativity, we know that the speed of light is a constant, c, for all reference frames. In the unprimed frame, from the definition of speed:

[math]\displaystyle{ speed=\frac{\Delta Distance}{\Delta Time} }[/math]

[math]\displaystyle{ c=\frac{\Delta d}{\Delta t} }[/math]

where

[math]\displaystyle{ c=3\times 10^8\ m/s }[/math]

Using the distance equation in a Cartesian coordinate system, the equation for the speed of light becomes

[math]\displaystyle{ c=\frac{\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}}{\Delta t} }[/math]

Following the postulate of Special Relativity, this implies for the primed frame

[math]\displaystyle{ c=\frac{\sqrt{(\Delta x')^2+(\Delta y')^2+(\Delta z')^2}}{\Delta t'} }[/math]

We can rewrite this as

[math]\displaystyle{ \frac{(\Delta x')^2+(\Delta y')^2+(\Delta z')^2}{(\Delta t')^2}= c^2=\frac{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}{(\Delta t)^2} }[/math]

This is possible since the ratios of distance to time are multiples of the same base, i.e. the square of the speed of light [math]\displaystyle{ (\frac{3\times 10^8\ m}{s})^2 }[/math]. Therefore for the relative change in the time in one frame, the distance must change by the same factor to maintain the same constant. With this we can write

[math]\displaystyle{ c^2 \Delta t^{'2}=(\Delta x')^2+(\Delta y')^2+(\Delta z')^2\ \ \ \ \ c^2 \Delta t^{2}=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2 }[/math]

[math]\displaystyle{ \Rightarrow c^2 \Delta t^{'2}-(\Delta x')^2-(\Delta y')^2-(\Delta z')^2= c^2 \Delta t^{2}-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2 }[/math]

This quantity is known as the time space interval [math]\displaystyle{ ds^2 }[/math] when the change is infinitesimal

[math]\displaystyle{ ds^2\equiv c^2 d t^{'2}-(d x')^2-(d y')^2-(d z')^2= c^2 d t^{2}-(d x)^2-(d y)^2-(d z)^2 }[/math]

Since the speed of light is a constant for all frames of reference, this allows the space time interval to also be invariant.

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